I'm sure it 'can' or does happen, but it doesn't mean it's an automatic thing, nor does it equate that it's happening because there's a bigfoot around.
We've experienced many episodes where it's not the woods going quiet, but the woods going nuts when there are BF around.
Without hard, physical evidence or scientifically obtained and vetted data, everything about this is purely conjecture.
Edited to Add: If it's an eyewitness who just had an encounter reporting this happening, then it's even more likely it's an adrenaline induced auditory event experienced by the individual. The temporary loss of auditory perception and logic/reasoning skills is a medically documented phenomenon in a shocking or frightening situation.